A non-Hermitian Hamilton operator and the physics of open quantum systems

Abstract

A powerful method for the description of open quantum systems is the Feshbach projection operator (FPO) technique. In this formalism, the whole function space is divided into two subspaces that are coupled with one another. One of the subspaces contains the wave functions localized in a certain finite region while the continuum of extended scattering wave functions is involved in the other subspace. The Hamilton operator of the whole system is Hermitian, that of the localized part is, however, non-Hermitian. This non-Hermitian Hamilton operator H eff represents the core of the FPO method in present-day studies. It gives a unified description of discrete and resonance states. Furthermore, it contains the time operator. The eigenvalues z λ and eigenfunctions φ λ of H eff are an important ingredient of the S matrix. They are energy dependent. The phases of the φ λ are, generally, nonrigid. Most interesting physical effects are caused by the branch points in the complex plane. On the one hand, they cause the avoided level crossings that appear as level repulsion or widths bifurcation in approaching the branch points under different conditions. On the other hand, observable values are usually enhanced and accelerated in the vicinity of the branch points. In most cases, the theory is time asymmetric. An exception are the PT symmetric bound states in the continuum appearing in space symmetric systems due to the avoided level crossing phenomenon in the complex plane. In the paper, the peculiarities of the FPO method are considered and three typical phenomena are sketched: (i) the unified description of decay and scattering processes, (ii) the appearance of bound states in the continuum and (iii) the spectroscopic reordering processes characteristic of the regime with overlapping resonances.

Coalescence of exceptional points and phase diagrams for one-dimensionalPT-symmetric photonic crystals

“…To identify the order of singularity for points S, M, R, and N, we study the phase rigidity r j for bands j = 2, 3, and 4 defined as r j (k) = ũ R jk |ũ R jk −1 , where |ũ R jk are the normalized right eigenstates [41]. Phase rigidity measures the amount of mixing of two states near an EP.…”

“…6(a) and 6(c), respectively. In general, bands can be entangled and the non-Abelian Zak phase can be calculated by using the following bilinear product [41,46]:…”

Resolubility of image-potential resonances

“…In the literature, the effective Hamilton operator H eff for the general case of n levels interacting with several continua is usually obtained by means of the Feshbach projection operator formalism [22,24,37]. Much of the essential physics, however, is already revealed by considering two levels coupled to one structureless continuum [38].…”

Noise-assisted quantum electron transfer in photosynthetic complexes

“…1.) Using the standard Feshbach projection method [24][25][26][27][28], one can show that the dynamics of the donoracceptor (intrinsic) states can be described by the follow- ing Schrödinger equation with an effective non-Hermitian Hamiltonian,H = H − iW (we set = 1) [29]:…”

“…7). Consequently, the condition of validaty of the approximation (26) - (27) can be written as V ≪ Dσ. This rough estimate can be improved greatly for the high level of noise leading to V ≤ Dσ.…”